Question

1. $x + 7y = -15$

$10x + 14y = 18$

1. $-6x + 3y = 12$

$12x - 6y = -24$

Explanation:

Problem 2: Solve the system of equations $\boldsymbol{x + 7y = -15}$ and $\boldsymbol{10x + 14y = 18}$

Step 1: Simplify the second equation

Divide the second equation $10x + 14y = 18$ by 2 to simplify.
$$\frac{10x + 14y}{2}=\frac{18}{2}$$
$$5x + 7y = 9$$

Step 2: Subtract the first equation from the new second equation

We have the first equation $x + 7y = -15$ and the new second equation $5x + 7y = 9$. Subtract the first from the second to eliminate $y$.
$$(5x + 7y)-(x + 7y)=9 - (-15)$$
$$5x + 7y - x - 7y=9 + 15$$
$$4x=24$$

Step 3: Solve for $x$

Divide both sides of $4x = 24$ by 4.
$$x=\frac{24}{4}$$
$$x = 6$$

Step 4: Substitute $x = 6$ into the first equation to find $y$

Substitute $x = 6$ into $x + 7y = -15$.
$$6 + 7y=-15$$
Subtract 6 from both sides:
$$7y=-15 - 6$$
$$7y=-21$$
Divide by 7:
$$y=\frac{-21}{7}$$
$$y=-3$$

Step 1: Simplify the first equation

Divide the first equation $-6x + 3y = 12$ by 3.
$$\frac{-6x + 3y}{3}=\frac{12}{3}$$
$$-2x + y = 4$$
We can rewrite this as $y=2x + 4$

Step 2: Substitute $y = 2x + 4$ into the second equation

Substitute $y = 2x + 4$ into $12x - 6y = -24$.
$$12x-6(2x + 4)=-24$$
Expand the left side:
$$12x-12x - 24=-24$$
Simplify:
$$-24=-24$$

This is a true statement, which means the two equations are dependent (they represent the same line) and there are infinitely many solutions. We can express the solution in terms of $x$: $y = 2x + 4$, where $x$ can be any real number.

Answer:

(Problem 2):
The solution is $x = 6$, $y = -3$

Problem 4: Solve the system of equations $\boldsymbol{-6x + 3y = 12}$ and $\boldsymbol{12x - 6y = -24}$